This pages lists the courses in Number Theory that are taught on a regular basis at the University of Pisa. For each course we give a brief summary in English, a more detailed description of the content in Italian ("Programma" and "Registro lezioni"), and links to lecture notes (as taken by students), when they are available.
- Teoria dei Campi e Teoria di Galois (Galois Theory)
An introduction to Galois theory aimed at third- and fourth-year students. Topics covered include infinite Galois extensions, Kummer theory, ...
Programma Registro delle lezioni (2015-2016) Notes by V. Galgano and C. Sircana Notes by G. Mezzedimi - Teoria Algebrica dei Numeri 1 (Algebraic Number Theory 1)
A first course in Algebraic Number Theory, covering the notion of number field, the splitting of primes in finite extensions of $\mathbb{Q}$, finiteness of the class group, and Dirichlet's unit theorem.
Programma Registro delle lezioni (2014-2015) Notes by G. Mezzedimi - Teoria Algebrica dei Numeri 2 (Algebraic Number Theory 2)
An introduction to the theory of $p$-adic fields: Ostrowski's theorem on valuations of $\mathbb{Q}$, inertia, ramification and ramification groups, Krasner's lemma, different and discriminant, and an introduction to local class field theory.
Programma Registro delle lezioni (2015-2016) Notes by G. Mezzedimi and C. Sircana - Curve Algebriche (Algebraic Curves)
This is a first course on elliptic curves. Besides the basic definitions, topics covered include the notion of good and bad reduction, isogenies, the Tate module of an elliptic curve, and the Mordell-Weil theorem.
Programma (missing) Registro delle lezioni (2016-2017) - Teoria dei Numeri Elementare (Elementary Number Theory)
This course covers a number of classical topics in elementary number theory (the Chinese Remainder Theorem, higher-degree congruences modulo prime powers, Legendre and Jacobi symbols, primitive roots, Moebius inversion, ...), as well as some more analytic subjects, such as the theorems of Mertens and Chebyshev. A final section of the course is devoted to a single more advanced topic (some examples from past years: Schnirelmann density; the elementary proof of the Prime Number Theorem; ...)
Programma (missing) Registro delle lezioni (2015-2016) - Teoria Analitica dei Numeri A (Analytic number theory A)
This course covers many classical topics in analytic number theory, such as the theory of Dirichlet characters and of their associated $L$-functions, as well as many properties of Riemann's $\zeta$ function and Euler's $\Gamma$. The Siegel-Walfisz and Bombieri-Vinogradov theorems are also touched upon.
Programma (missing) Registro delle lezioni (2016-2017) - Funzioni speciali (Special Functions) Programma (missing) Registro delle lezioni (2015-2016, missing)